Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6,8,9)(7,11,12,10)(14,18)(15,21)(20,25)(22,27)(32,39)(35,42)(37,43)(40,44);(6,9)(10,11)(15,21)(19,24)(22,27)(26,28)(29,30,33,36)(31,34,38,41)(32,35,39,42)(37,40,43,44);(5,7,8,12)(6,10,9,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,31,33,38)(30,34,36,41)(32,37,39,43)(35,40,42,44);(5,7)(6,10)(8,12)(9,11)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32)(30,35)(31,37)(33,39)(34,40)(36,42)(38,43)(41,44)Polycyclic presentation:
2^9|a^2=fgi,b^2=h,c^2=f,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4516521);